Hi mr.Al and many thanks !
So everytime i must solve the transfer function to learn what is the fcut?
The wikipedia is wrong so?
Hello again,
Well if you want to do an unknown circuit then you have to solve it which means evaluating the transfer function, or else use ready made software. When you evaluate it you can find the formula that explicitly solves for a given thing like fc.
Wikipedia is not wrong, but they are not giving the information that is usually desired by a designer. Namely, they are not providing a formula for fc (or wc) they are providing a formula for f0 (or w0), which is not the same thing. So you can not use their formula to find the cutoff frequency without taking additional steps. The formulas i provided earlier calculate wc directly from which you can easily get fc.
The Texas Instruments paper does the same thing. They calculate some things explicitly and others im not even sure if they calculate at all, and these are things that they take for granted in the formulas. For example, FSF, they dont seem to provide an explicit calculation for this do they? If we are to use w0 or f0 from Wikipedia then we also need to know how to calculate FSF.
Note also that in the example for the low pass Butterworth Sallen-Key they coincidentally use an FSF=1, which to me says they are cheating on their own paper.
So if you want simple formulas for known circuits that's one thing, but if you want to know how to solve every circuit like this even with different topologies, then you should learn how to calculate the transfer function one way or another.
I went ahead and calculated the FSF for any 2nd order filter with passband gain equal to 1. The result is:
FSF=sqrt(sqrt(2)*sqrt(2*d^4-2*d^2+1)-2*d^2+1)
[LATEX]FSF=\sqrt{\sqrt{2}\,\sqrt{2\,{d}^{4}-2\,{d}^{2}+1}-2\,{d}^{2}+1}[/LATEX]
where d is the damping factor. This means if you calculate w0 from the Wikipedia site you can then calculate wc from:
wc=FSF*w0
or:
fc=FSF*f0
The damping factor is calculated from:
d=(R1+R2)/(2*w0*C1*R1*R2)
and again w0=1/sqrt(R1*R2*C1*C2)
and just to note, when R2=R1 and C2=C1 the damping factor is exactly equal to 1, but FSF is not necessarily equal to 1. In fact for the original circuit with original values 10k and 1nf, we get:
FSF=0.64359425290558
and multiplying this by 15.9kHz we get 10.2kHz which is the right value for the cutoff frequency fc.
We might also note that the only time we get FSF=1 is when the damping factor d is equal to 1/sqrt(2).
Also note that the FSF given here is the reciprocal of the FSF given in the Texas Instruments paper.